2 0 1 1 5 May 2 0 1 1 4 0 3 Journal of Shanghai Normal University Philosophy & Social Sciences Edition Vol. 40 No. 3 F830. 91 A 1004-8634 2011 03-0032- 12 Pair Copula - GARCH - t 200234 Pair copula - garch - t C D Pair copula - GARCH - t C D 2010 6 2010 11 9 2. 49% 2007 6. 9% 2007 9 2010 6 2010 11 1. 93% 9. 60% 6. 02% 2006 8 1 2010 10 25 5 2011-01-12 copula A - 3138-11 - 020006 2009BJB022 Copula - GARCH - VaR 09ZS142 1975-32
Pair Copula - GARCH - t 5 Domowitz and Hakkio 1985 ARCH GARCH Nieuwland et al. 1998 1986 1991 GARCH BF / DM DG /DM FF /DM IL /DM SP /DM 3 6 12 Torben Bollerslev 2003 GARCH Kenneth and Dongchul 1995 GARCH GARCH Christopher and Weller 2001 GARCH 2003 GARCH GARCH GARCH 2005 / ARIMA GARCH GARCH / 2007 2007 GARCH 33
3 2011 Patton 2006 GARCH copula copula Chen and Fan 2006 copula DCC Engle 2002 copula GARCH copula GARCH Czado 2010 ARAM - GARCH copula copula Regular copula copula Canonical copula GARCH - t DCC copula Pair copula - GARCH - t 2002 copula 2006 GARCH - t copula copula - GARCH 2009 copula 2011 copula vines copula - ARMA - GARCH - t pair copula C D 5 copula copula copula 1. Copula - ARMA 1 1 - GARCH 1 - t ARMA 1 - GARCH 1 GARCH - t X it GARCH - t ARMA - GARCH GARCH - t coupla copula - AR- MA - GARCH - t X it 2 σ i t = i X i t -1 + σ i = ω i + α i ε 2 i t -1 t Z i t + θ i Z i t -1 2 + β i σ i t -1 ε i t = σ i t E Z i t = 0 Var Z i t = 1 ξ 1t ξ 2t ξ it ~ C t T v1 ξ 1t T v2 ξ 2t T vi ξ it i = 1 2 5. Z i t 0 1 i θ i i = 1 2 5 ω i α i β i i = 1 2 5 C t.. i copula T vi. 0 1 v i t 1 v n 槡 v n - 2 ξ it ~ t v i i = 1 2 5 i = 2 5 34
Pair Copula - GARCH - t 2. Pair - copula USD /RMB EUR /RMB JPY /RMB HKD /RMB POUND /RMB r i i = 1 2 5 r 1 r 5 F f f r 1 r 5 = f r 5 r 1 r 4 f r 1 r 4 = = Π 5 f r i r 1 r i -1 f r 1. 2 i = 2 F.. f.. Skalar f r 5 r 1 r 4 f r 5 r 1 r 4 = f r 4 r 5 r 1 r 2 r 3 f r 4 r 1 r 2 r 3 = c 4 5 1 2 3 f r 4 r 1 r 2 r 3. c.. copula i ji < j i 1 < < i k c i j i1 i k = c i j i1 i k F r i r i1 r ik F r j r i1 r ik. 3 2 f r 1 r 5 = f r 1 Π 5-1 i = 2k Πi = 1c i -k i 1 i -k -1 = Π 5 u = 1 Bedford and Cooke f r u Π 5-1 i = 2k Πi = 1c i -k i 1 i -k -1 f x i 3 4 = Π 5 f r u Π 4 -j u = 1 j = 1m Π5 = 1c j j+m 1 j-1 j = i - k j + m = i. 2001 2002 4 Pair copula coupla copula copula copula copula i i - 1 i - 1 copula i - 2 copula copula copula C copula D 5 C D 1 1 Pair copula c F r 1 F r 5 = c F r 1 F r 5 = c 12 c 13 c 14 c 15 c 12 c 23 c 34 c 45 tree1 c 23 1 c 24 1 c 25 1 c 13 2 c 24 3 c 35 4 tree2 c 35 12 c 34 12 c 14 23 c 25 34 tree3 c 45 12 c 15 23 tree4 copula C D C ij kl c F r i kl G r j kl F r i kl G r j kl r i kl r j kl 3. pair copula Marshal and Zeevi 2002 DCC R t DCC 1 1 Q t = 1 - α-β 珚 Q + αε t -1 ε t -1 + β Q t -1 R t = 珟 Q -1 t Q t 珟 Q -1 t. 5 α β 0 1 α + β < 1 珚 Q ε t 珟 Q t Q t p p DCC 1 1 TVC Patton 35
3 2011 Pair copula - GARCH - t 1027 Pair copula copula clayton Joe - Clayton copula JCC Clayton copula C cl u v θ = u -θ + v -θ - 1-1 /θ C cl u v θ = 1 + θ uv -θ-1 u -θ + v -θ - 1-2 -1 /θ. 6 θ 0 JCC C JCC u v κ γ = 1-1 - 1 - u κ -γ + 1-1 - v κ -γ - 1-1 /γ. 7 κ 1 γ > 0 κ γ JCC τ l = 2-1 /γ τ u = 2-2 1 /κ. 8 γ κ JCC 2006 8 1 2010 10 25 5140 matlab2010b XR t r t r t = log XR t - log XR t -1. 5 2 2 Mean Std Skewness Kurtosis KS P USD - 1. 526e - 4 0. 0081-0. 6793 7. 0625 < 1. 0000e - 3 EUR - 5. 480e - 5 0. 0069-0. 5171 12. 6482 < 1. 0000e - 3 HKD 1. 548e - 4 0. 0009-0. 6623 7. 1352 < 1. 0000e - 3 JPY 9. 230e - 5 0. 0072 0. 0523 6. 3267 < 1. 0000e - 3 Pound - 3. 428e - 4 0. 0073-0. 4317 7. 5324 < 1. 0000e - 3 2 5 0 3 5 KS P 0. 001 ADF 3 R i R i i = 1 2 3 4 5 36
Pair Copula - GARCH - t 3 ADF ADF 5% R i R i 1% Dollar EURO HK $ Yen Pound - 6. 8508-25. 2884-2. 5681-1. 9413 4. 1251-26. 9443-3. 4391-2. 8653-0. 2810-27. 6632-3. 9707-3. 4160 0. 5457-27. 5600-2. 5681-1. 9413-1. 3561-27. 5600-3. 4391-2. 8653-3. 9685-27. 5425-3. 9707-3. 4160-0. 9187-26. 1026-2. 5681-1. 9413-2. 3238-26. 1050-3. 4391-2. 8653-2. 6192-26. 0892-3. 9707-3. 4160-6. 6106-26. 9745-2. 5681-1. 9413 3. 5262-28. 6138-3. 4391-2. 8653-0. 3497-29. 3174-3. 9707-3. 4160-0. 9221-22. 7164-2. 5698-1. 9415-0. 2846-22. 7386-3. 4434-2. 8674-1. 800-22. 8025-3. 9775-3. 4192 3 ADF ADF 1% 5% ADF ADF 1% 5% ADF 5% ADF 1 1. ARMA 1 1 - GARCH 1 1 - t 4 4 ARMA 1 1 - GARCH - t ARMA 1 C USD /RMB EUR /RMB JPY /RMB HKD /RMB POUND /RMB - 2. 9775e - 005 1. 3413e - 003-2. 4551e - 004-6. 6083e - 005-7. 7378e - 005 2. 4420e - 5 2. 7590e - 004 3. 2989e - 004 2. 9736e - 005 1. 5729e - 004-1. 2293 0. 4862-0. 7442-2. 2223-0. 4920-0. 6152-0. 7732-0. 7218-0. 7180 0. 4976 AR 1 0. 1461 0. 2866 0. 0025 0. 0096 0. 8619-4. 2210 0. 4862 398. 117-74. 9347 0. 5780 0. 6871 0. 7423-0. 7285 0. 7423-0. 4738 MA 1 0. 1321 0. 3045 0. 2667 0. 0010 0. 8744 5. 2009 2. 4463-2. 7065 74. 2559-0. 5418 GARCH 1 1 - t K 7. 5801e - 008 2e - 007 6. 3977e - 007-6. 6083e - 005 2. 3679e - 007 1. 2247e - 08 1. 2741e - 007 2. 9482e - 007 1. 7155e - 008 1. 4595e - 007 6. 1893 1. 5697 2. 1700 4. 8536 1. 6224 0. 5068 0. 9484 0. 9235 0. 5625 0. 9342 GARCH 1 0. 0541 0. 0121 0. 0161 0. 0572 0. 0160 9. 3690 78. 6045 55. 5809 9. 8384 58. 4628 37
3 2011 ARCH 1 LEVERAGE 1 DOF USD /RMB EUR /RMB JPY /RMB HKD /RMB POUND /RMB 0. 2800 0. 0954 4. 3648 0. 4267 0. 1310 3. 2576 2. 8314 0. 0562 2. 9342 0. 0366 0. 0167 2. 1947 0. 0229 0. 2000 1. 1422 7. 6492 1. 1742 6. 5145 0. 1177 0. 0235 4. 9978-0. 1047 0. 0283-3. 6946 6. 1651 1. 3910 4. 4322 0. 2703 0. 0828 3. 2662 0. 3345 0. 1312 2. 5391 3. 2365 0. 0250 129. 4170 0. 0460 0. 0216 2. 1071 0. 0329 0. 0240 1. 3706 9. 3822 0. 0072 1298. 7311 log L 6. 1633e + 03 3. 8214e + 003 3. 3672e + 003 6. 020e + 3 3. 7829e + 3 AIC - 1. 2311e - 004-7. 6269e + 003-7. 3287e + 003-1. 2026e + 004-7. 5498e + 003 BIC - 1. 2271e - 004-7. 5874e + 003-7. 2893e + 003-1. 1986 + e004-7. 5103e + 003 t DOF Log L AIC BIC ARMA P Q - GARCH p q LB ARMA 1 1 - GARCH 1 1 1 Q - Q GARCH - t copula ARMA 1 1 - GARCH 1 1 - t ε t t - 1 T ε i ~ F i = 1 Tu t = F ε t ε t u i ~ U 0 1 i = 1 T F i x = 1 T + 1 j T 1 Xij x 1 = 1 1 = { 1 T 0 2. ARMA 1 1 - GARCH 1 1 copula - t 5 ARMA 1 1 - GARCH 1 - t U 0 1 Q - Q 2 USD /RMB EUR /RMB USD /RMB HKD /RMB copula - t copula - t 0 1 kendall τ ρ = sin πτ /2 ρ copula - t 5 copula - t USD /RMB JPY /RMB HKD /RMB POUND /RMB EUR /RMB 8. 8925 3. 6475 7. 4049 13. 8329 148. 8927 32. 4568 82. 3105 244. 3721 USD /RMB 12. 0586 2. 0276 16. 0376 19. 1908 853. 1859 44. 2544 JPY /RMB 13. 0006 4. 0078 12. 8482 22. 9098 HKD /RMB 10. 3594 23. 7544 copula - t 38
Pair Copula - GARCH - t 1 Q - Q 2 5 5 USD / RMB JPY /RMB 12. 0586USD /RMB HKD /RMB 2. 0276 copula pair copula 6 copula - t EUR /RMB USD /RMB HKD /RMB EUR /RMB JPY /RMB POUND /RMB USD /RMB JPY /RMB 39
3 2011 6 copula - t USD /RMB JPY /RMB HKD /RMB POUND /RMB EUR /RMB - 0. 5385 0. 1104-0. 4033 0. 6186 USD /RMB - 0. 2176 0. 8922-0. 3186 JPY /RMB - 0. 1640-0. 0570 HKD /RMB - 0. 2297 6 USD /RMB HKD /RMB EUR /RMB POUND /RMB 0. 8922 0. 6186 copula - t pair copula 3. Pair copula Pair coupla 7 8 t v t 7 C copula USD / RMB HKD /RMB copula ν USD /RMB EUR /RMB JPY /RMB HKD /RMB POUND /RMB 3 7 Pair copula C copula t copula Clayton copula JOE - Clayton copula 9. 4512 C 12 0. 260 12. 2357 0. 0001 C 13 0. 478 0. 014 2. 1323 0. 6324 C 14 0. 004 0. 014 22. 4275 0. 0002 C 15 1. 099 0. 012 4. 2823 0. 0562 C 23 1 0. 161 0. 020 17. 2962 0. 0332 C 24 1 0. 508 0. 012 17. 5309 0. 3130 C 25 1 1. 779 0. 012 24. 8709 0. 0001 C 34 12 18. 035 0. 015 10. 7606 0. 0001 C 35 12 1. 753 0. 018 43. 9585 0. 0004 C 45 123 52. 108 0. 009 0. 6992 0. 024 0. 4496 0. 038 0. 8500 0. 040 0. 3470 0. 035 0. 0396 0. 065 AIC BIC LL - 2619. 5739 2570. 2300 1319. 787-1731. 1945-1681. 8506 875. 597-1976. 1442-1877. 4562 1008. 072 1 5 USD /RMB POUND /RMB 40
Pair Copula - GARCH - t Clayton copula 7 C copula USD /RMB EUR /RMB POUND /RMB θ USD /RMB HKD /RMB 0. 6324 copula - t 0. 8922 C USD /RMB USD /RMB EUR /RMB POUND /RMB 0. 3130 JOE - Clayton copula 7 C copula USD /RMB JPY /RMB USD / RMB POUND /RMB 0. 6992 0. 4496 EUR /RMB POUND /RMB USD / RMB 0. 8500 8 Pair copula C D D t copula Clayton copula JOE - Clayton copula t copula ν D copula copula USD /RMB HKD /RMB EUR /RMB JPY /RMB C Clayton copula USD /RMB EUR / RMB JPY /RMB HKD /RMB 0EUR /RMB JPY /RMB HKD /RMB POUND / RMB USD /RMB HKD /RMB EUR /RMB JPY /RMB 0. 6375 EUR /RMB POUND /RMB JPY /RMB HKD /RMB 0. 3135 3 EUR /RMB JPY /RMB HKD /RMB USD /RMB POUND /RMB JOE - Clayton copula C D EUR /RMB JPY /RMB JPY /RMB HKD / RMB EUR /RMB USD /RMB JPY /RMB 0. 4506 JPY /RMB HKD /RMB EUR /RMB POUND /RMB 0. 7880 8 Pair copula D copula t copula Clayton copula JOE - Clayton copula C 12 7. 3279 0. 088 0. 0173 0. 012 C 23 5. 0895 0. 029 0. 0619 0. 014 C 34 7. 2995 0. 201 C 45 15. 4637 0. 402 0. 0113 0. 016 0. 6946 0. 006 C 13 2 11. 5659 9. 939 0. 4506 0. 029 41
3 2011 copula t copula Clayton copula JOE - Clayton copula C 24 3 6. 2967 0. 054 0. 0269 0. 007 C 35 4 6. 3112 112. 226 C 14 23 3. 2078 64. 133 0. 6375 0. 013 C 25 34 19. 4615 222. 407 0. 3135 0. 015 0. 7880 0. 007 C 15 234 161. 8391 13. 149 0. 3592 0. 036 AIC - 2566. 3632-1760. 0770-2057. 4085 BIC - 2517. 0192-1710. 7330-1958. 7206 LL 1293. 182 890. 039 1048. 704 1 5 USD /RMB POUND /RMB Pair copula - GARCH - t C D 5 DCC 1 1 5 C t copula USD /RMB HKD /RMB EUR /RMB POUND /RMB 1 Clayton copula USD /RMB USD /RMB HKD /RMB 0 JCC D t copula C Clayton copula EUR /RMB JPY /RMB USD /RMB HKD / RMB JPY /RMB HKD /RMB EUR /RMB POUND /RMB JOE - Clayton copula 4 SV 42
Pair Copula - GARCH - t copula 1. J. 2005 4. 2. GARCH J. 2003 5. 3. J. 2007 2. 4. J. 2007 10. 5. copula vines J. 2011 6. 6. copula J. 2009 1. 7. Copula - GARCH J. 2010 5. 8 Aguilar J Nydahl S. Central Bank Intervention and Exchange Rates the Case of Sweden J. Journal of International Financial Markets Institutions and Money 2000 10. 9 Beine M Charles S Sebastien Laurent. The Impact of Central Bank FX Interventions on Currency Components J. Journal of Financial E- conometrics 2007 1. 10 Christopher J Neely Paul A Weller. Predicting Exchange Rate Volatility Genetic Programming vs. GARCH and RiskMetrics Workingpaper Federal reserve bank of St M. Louis 2001. 11 Domowitz I Hakkio C S. Conditional Variance and the Risk Premium in the Foreign Exchange Market J. Journal of International Economics 1985 19. 12 Niwuwland F G M C Verschoor W F C Wolff C C P. EMS Exchange Rate Expectations and Time - varying Risk Premia J. Economics Letters 1998 3. 13 Torben G Andersen Tim Bollerslev Francis X Diebold Paul Labys. Modeling and Forecasting Realized Volatility J. Journal of Econometrica Econometric Society 2003 2. 14 West Kenneth D Hali J Edison Dongchul Cho. A Utility Based Comparison of Some Models of Foreign Exchange Volatility J. Journal of International Economics 1993 4. 15 Nelsen R B. An introduction to copula 2 nd second. Springer series in statistics Z. Springer New York 2006. 16 Patton A J. Estimation of multivariate models for time series of possibly different lengths J. Journal of applied econometrics 21 2006. 17 Aas K Czado C Frigessi A Bakken H. Pair - copula constructions of multiple dependence J. Insurance mathematics and economics 2009 44. 18 Czado C Aleksey M Tanja B Rada D. manuscript 2010. Pair - copula constructions for modeling exchange rate dependce Z. Unpublished A Positive Analysis of RMB Exchange Rates Based on the Pair Copula - GARCH - t Model CUI Baisheng College of Finance Shanghai Normal University Shanghai 200234 China Abstract By building up the model called pair copula - GARCH - t this paper studies the correlation of the RMB against the US dollar the EURO the Hang Kong dollar the Yen and the Pound. It allows the conditional correlation among exchange rates to be both time - varying and modeled independently from the marginal distributions. Empirical evidence indicates that there is a significantly unconditional positive correlation between the exchange rates of USD / RMB and HKD / RMB in the canonical copula vine. In the D - copula vine there is no significantly unconditional correlation. Key words RMB Exchange Rate Pair Copula - GARCH - t Model Canonical Copula D - copula volatility 43